Integrand size = 23, antiderivative size = 104 \[ \int \frac {\cosh ^5(c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^2} \, dx=-\frac {\left (3 a^2-2 a b-b^2\right ) \arctan \left (\frac {\sqrt {b} \sinh (c+d x)}{\sqrt {a}}\right )}{2 a^{3/2} b^{5/2} d}+\frac {\sinh (c+d x)}{b^2 d}+\frac {(a-b)^2 \sinh (c+d x)}{2 a b^2 d \left (a+b \sinh ^2(c+d x)\right )} \] Output:
-1/2*(3*a^2-2*a*b-b^2)*arctan(b^(1/2)*sinh(d*x+c)/a^(1/2))/a^(3/2)/b^(5/2) /d+sinh(d*x+c)/b^2/d+1/2*(a-b)^2*sinh(d*x+c)/a/b^2/d/(a+b*sinh(d*x+c)^2)
Time = 0.23 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.02 \[ \int \frac {\cosh ^5(c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^2} \, dx=\frac {-\frac {\left (-3 a^2+2 a b+b^2\right ) \arctan \left (\frac {\sqrt {a} \text {csch}(c+d x)}{\sqrt {b}}\right )}{a^{3/2}}+2 \sqrt {b} \sinh (c+d x)+\frac {2 (a-b)^2 \sqrt {b} \sinh (c+d x)}{a (2 a-b+b \cosh (2 (c+d x)))}}{2 b^{5/2} d} \] Input:
Integrate[Cosh[c + d*x]^5/(a + b*Sinh[c + d*x]^2)^2,x]
Output:
(-(((-3*a^2 + 2*a*b + b^2)*ArcTan[(Sqrt[a]*Csch[c + d*x])/Sqrt[b]])/a^(3/2 )) + 2*Sqrt[b]*Sinh[c + d*x] + (2*(a - b)^2*Sqrt[b]*Sinh[c + d*x])/(a*(2*a - b + b*Cosh[2*(c + d*x)])))/(2*b^(5/2)*d)
Time = 0.36 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.95, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3042, 3669, 300, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\cosh ^5(c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\cos (i c+i d x)^5}{\left (a-b \sin (i c+i d x)^2\right )^2}dx\) |
\(\Big \downarrow \) 3669 |
\(\displaystyle \frac {\int \frac {\left (\sinh ^2(c+d x)+1\right )^2}{\left (b \sinh ^2(c+d x)+a\right )^2}d\sinh (c+d x)}{d}\) |
\(\Big \downarrow \) 300 |
\(\displaystyle \frac {\int \left (\frac {1}{b^2}-\frac {a^2-b^2+2 (a-b) b \sinh ^2(c+d x)}{b^2 \left (b \sinh ^2(c+d x)+a\right )^2}\right )d\sinh (c+d x)}{d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {-\frac {\left (3 a^2-2 a b-b^2\right ) \arctan \left (\frac {\sqrt {b} \sinh (c+d x)}{\sqrt {a}}\right )}{2 a^{3/2} b^{5/2}}+\frac {(a-b)^2 \sinh (c+d x)}{2 a b^2 \left (a+b \sinh ^2(c+d x)\right )}+\frac {\sinh (c+d x)}{b^2}}{d}\) |
Input:
Int[Cosh[c + d*x]^5/(a + b*Sinh[c + d*x]^2)^2,x]
Output:
(-1/2*((3*a^2 - 2*a*b - b^2)*ArcTan[(Sqrt[b]*Sinh[c + d*x])/Sqrt[a]])/(a^( 3/2)*b^(5/2)) + Sinh[c + d*x]/b^2 + ((a - b)^2*Sinh[c + d*x])/(2*a*b^2*(a + b*Sinh[c + d*x]^2)))/d
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Int [PolynomialDivide[(a + b*x^2)^p, (c + d*x^2)^(-q), x], x] /; FreeQ[{a, b, c , d}, x] && NeQ[b*c - a*d, 0] && IGtQ[p, 0] && ILtQ[q, 0] && GeQ[p, -q]
Int[cos[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^ (p_.), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Simp[ff/f S ubst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b*ff^2*x^2)^p, x], x, Sin[e + f*x] /ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]
Time = 143.92 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.93
method | result | size |
derivativedivides | \(\frac {\frac {\sinh \left (d x +c \right )}{b^{2}}-\frac {-\frac {\left (a^{2}-2 a b +b^{2}\right ) \sinh \left (d x +c \right )}{2 a \left (a +b \sinh \left (d x +c \right )^{2}\right )}+\frac {\left (3 a^{2}-2 a b -b^{2}\right ) \arctan \left (\frac {b \sinh \left (d x +c \right )}{\sqrt {a b}}\right )}{2 a \sqrt {a b}}}{b^{2}}}{d}\) | \(97\) |
default | \(\frac {\frac {\sinh \left (d x +c \right )}{b^{2}}-\frac {-\frac {\left (a^{2}-2 a b +b^{2}\right ) \sinh \left (d x +c \right )}{2 a \left (a +b \sinh \left (d x +c \right )^{2}\right )}+\frac {\left (3 a^{2}-2 a b -b^{2}\right ) \arctan \left (\frac {b \sinh \left (d x +c \right )}{\sqrt {a b}}\right )}{2 a \sqrt {a b}}}{b^{2}}}{d}\) | \(97\) |
risch | \(\frac {{\mathrm e}^{d x +c}}{2 b^{2} d}-\frac {{\mathrm e}^{-d x -c}}{2 b^{2} d}+\frac {\left (a^{2}-2 a b +b^{2}\right ) {\mathrm e}^{d x +c} \left ({\mathrm e}^{2 d x +2 c}-1\right )}{d \,b^{2} a \left (b \,{\mathrm e}^{4 d x +4 c}+4 \,{\mathrm e}^{2 d x +2 c} a -2 \,{\mathrm e}^{2 d x +2 c} b +b \right )}-\frac {3 a \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \,{\mathrm e}^{d x +c}}{\sqrt {-a b}}-1\right )}{4 \sqrt {-a b}\, d \,b^{2}}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \,{\mathrm e}^{d x +c}}{\sqrt {-a b}}-1\right )}{2 \sqrt {-a b}\, d b}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \,{\mathrm e}^{d x +c}}{\sqrt {-a b}}-1\right )}{4 \sqrt {-a b}\, d a}+\frac {3 a \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 a \,{\mathrm e}^{d x +c}}{\sqrt {-a b}}-1\right )}{4 \sqrt {-a b}\, d \,b^{2}}-\frac {\ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 a \,{\mathrm e}^{d x +c}}{\sqrt {-a b}}-1\right )}{2 \sqrt {-a b}\, d b}-\frac {\ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 a \,{\mathrm e}^{d x +c}}{\sqrt {-a b}}-1\right )}{4 \sqrt {-a b}\, d a}\) | \(358\) |
Input:
int(cosh(d*x+c)^5/(a+b*sinh(d*x+c)^2)^2,x,method=_RETURNVERBOSE)
Output:
1/d*(1/b^2*sinh(d*x+c)-1/b^2*(-1/2*(a^2-2*a*b+b^2)/a*sinh(d*x+c)/(a+b*sinh (d*x+c)^2)+1/2*(3*a^2-2*a*b-b^2)/a/(a*b)^(1/2)*arctan(b*sinh(d*x+c)/(a*b)^ (1/2))))
Leaf count of result is larger than twice the leaf count of optimal. 1444 vs. \(2 (92) = 184\).
Time = 0.13 (sec) , antiderivative size = 2741, normalized size of antiderivative = 26.36 \[ \int \frac {\cosh ^5(c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^2} \, dx=\text {Too large to display} \] Input:
integrate(cosh(d*x+c)^5/(a+b*sinh(d*x+c)^2)^2,x, algorithm="fricas")
Output:
[1/4*(2*a^2*b^2*cosh(d*x + c)^6 + 12*a^2*b^2*cosh(d*x + c)*sinh(d*x + c)^5 + 2*a^2*b^2*sinh(d*x + c)^6 + 2*(6*a^3*b - 7*a^2*b^2 + 2*a*b^3)*cosh(d*x + c)^4 + 2*(15*a^2*b^2*cosh(d*x + c)^2 + 6*a^3*b - 7*a^2*b^2 + 2*a*b^3)*si nh(d*x + c)^4 - 2*a^2*b^2 + 8*(5*a^2*b^2*cosh(d*x + c)^3 + (6*a^3*b - 7*a^ 2*b^2 + 2*a*b^3)*cosh(d*x + c))*sinh(d*x + c)^3 - 2*(6*a^3*b - 7*a^2*b^2 + 2*a*b^3)*cosh(d*x + c)^2 + 2*(15*a^2*b^2*cosh(d*x + c)^4 - 6*a^3*b + 7*a^ 2*b^2 - 2*a*b^3 + 6*(6*a^3*b - 7*a^2*b^2 + 2*a*b^3)*cosh(d*x + c)^2)*sinh( d*x + c)^2 + ((3*a^2*b - 2*a*b^2 - b^3)*cosh(d*x + c)^5 + 5*(3*a^2*b - 2*a *b^2 - b^3)*cosh(d*x + c)*sinh(d*x + c)^4 + (3*a^2*b - 2*a*b^2 - b^3)*sinh (d*x + c)^5 + 2*(6*a^3 - 7*a^2*b + b^3)*cosh(d*x + c)^3 + 2*(6*a^3 - 7*a^2 *b + b^3 + 5*(3*a^2*b - 2*a*b^2 - b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^3 + 2*(5*(3*a^2*b - 2*a*b^2 - b^3)*cosh(d*x + c)^3 + 3*(6*a^3 - 7*a^2*b + b^3) *cosh(d*x + c))*sinh(d*x + c)^2 + (3*a^2*b - 2*a*b^2 - b^3)*cosh(d*x + c) + (5*(3*a^2*b - 2*a*b^2 - b^3)*cosh(d*x + c)^4 + 3*a^2*b - 2*a*b^2 - b^3 + 6*(6*a^3 - 7*a^2*b + b^3)*cosh(d*x + c)^2)*sinh(d*x + c))*sqrt(-a*b)*log( (b*cosh(d*x + c)^4 + 4*b*cosh(d*x + c)*sinh(d*x + c)^3 + b*sinh(d*x + c)^4 - 2*(2*a + b)*cosh(d*x + c)^2 + 2*(3*b*cosh(d*x + c)^2 - 2*a - b)*sinh(d* x + c)^2 + 4*(b*cosh(d*x + c)^3 - (2*a + b)*cosh(d*x + c))*sinh(d*x + c) - 4*(cosh(d*x + c)^3 + 3*cosh(d*x + c)*sinh(d*x + c)^2 + sinh(d*x + c)^3 + (3*cosh(d*x + c)^2 - 1)*sinh(d*x + c) - cosh(d*x + c))*sqrt(-a*b) + b)/...
Timed out. \[ \int \frac {\cosh ^5(c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^2} \, dx=\text {Timed out} \] Input:
integrate(cosh(d*x+c)**5/(a+b*sinh(d*x+c)**2)**2,x)
Output:
Timed out
\[ \int \frac {\cosh ^5(c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^2} \, dx=\int { \frac {\cosh \left (d x + c\right )^{5}}{{\left (b \sinh \left (d x + c\right )^{2} + a\right )}^{2}} \,d x } \] Input:
integrate(cosh(d*x+c)^5/(a+b*sinh(d*x+c)^2)^2,x, algorithm="maxima")
Output:
1/2*(a*b*e^(6*d*x + 6*c) - a*b + (6*a^2*e^(4*c) - 7*a*b*e^(4*c) + 2*b^2*e^ (4*c))*e^(4*d*x) - (6*a^2*e^(2*c) - 7*a*b*e^(2*c) + 2*b^2*e^(2*c))*e^(2*d* x))/(a*b^3*d*e^(5*d*x + 5*c) + a*b^3*d*e^(d*x + c) + 2*(2*a^2*b^2*d*e^(3*c ) - a*b^3*d*e^(3*c))*e^(3*d*x)) - 1/32*integrate(32*((3*a^2*e^(3*c) - 2*a* b*e^(3*c) - b^2*e^(3*c))*e^(3*d*x) + (3*a^2*e^c - 2*a*b*e^c - b^2*e^c)*e^( d*x))/(a*b^3*e^(4*d*x + 4*c) + a*b^3 + 2*(2*a^2*b^2*e^(2*c) - a*b^3*e^(2*c ))*e^(2*d*x)), x)
Exception generated. \[ \int \frac {\cosh ^5(c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^2} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(cosh(d*x+c)^5/(a+b*sinh(d*x+c)^2)^2,x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Limit: Max order reached or unable to make series expansion Error: Bad Argument Value
Timed out. \[ \int \frac {\cosh ^5(c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^2} \, dx=\int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^5}{{\left (b\,{\mathrm {sinh}\left (c+d\,x\right )}^2+a\right )}^2} \,d x \] Input:
int(cosh(c + d*x)^5/(a + b*sinh(c + d*x)^2)^2,x)
Output:
int(cosh(c + d*x)^5/(a + b*sinh(c + d*x)^2)^2, x)
Time = 0.21 (sec) , antiderivative size = 3876, normalized size of antiderivative = 37.27 \[ \int \frac {\cosh ^5(c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^2} \, dx =\text {Too large to display} \] Input:
int(cosh(d*x+c)^5/(a+b*sinh(d*x+c)^2)^2,x)
Output:
(6*e**(5*c + 5*d*x)*sqrt(b)*sqrt(a)*sqrt(a - b)*sqrt(2*sqrt(a)*sqrt(a - b) + 2*a - b)*atan((e**(c + d*x)*b)/(sqrt(b)*sqrt(2*sqrt(a)*sqrt(a - b) + 2* a - b)))*a**2*b - 4*e**(5*c + 5*d*x)*sqrt(b)*sqrt(a)*sqrt(a - b)*sqrt(2*sq rt(a)*sqrt(a - b) + 2*a - b)*atan((e**(c + d*x)*b)/(sqrt(b)*sqrt(2*sqrt(a) *sqrt(a - b) + 2*a - b)))*a*b**2 - 2*e**(5*c + 5*d*x)*sqrt(b)*sqrt(a)*sqrt (a - b)*sqrt(2*sqrt(a)*sqrt(a - b) + 2*a - b)*atan((e**(c + d*x)*b)/(sqrt( b)*sqrt(2*sqrt(a)*sqrt(a - b) + 2*a - b)))*b**3 + 24*e**(3*c + 3*d*x)*sqrt (b)*sqrt(a)*sqrt(a - b)*sqrt(2*sqrt(a)*sqrt(a - b) + 2*a - b)*atan((e**(c + d*x)*b)/(sqrt(b)*sqrt(2*sqrt(a)*sqrt(a - b) + 2*a - b)))*a**3 - 28*e**(3 *c + 3*d*x)*sqrt(b)*sqrt(a)*sqrt(a - b)*sqrt(2*sqrt(a)*sqrt(a - b) + 2*a - b)*atan((e**(c + d*x)*b)/(sqrt(b)*sqrt(2*sqrt(a)*sqrt(a - b) + 2*a - b))) *a**2*b + 4*e**(3*c + 3*d*x)*sqrt(b)*sqrt(a)*sqrt(a - b)*sqrt(2*sqrt(a)*sq rt(a - b) + 2*a - b)*atan((e**(c + d*x)*b)/(sqrt(b)*sqrt(2*sqrt(a)*sqrt(a - b) + 2*a - b)))*b**3 + 6*e**(c + d*x)*sqrt(b)*sqrt(a)*sqrt(a - b)*sqrt(2 *sqrt(a)*sqrt(a - b) + 2*a - b)*atan((e**(c + d*x)*b)/(sqrt(b)*sqrt(2*sqrt (a)*sqrt(a - b) + 2*a - b)))*a**2*b - 4*e**(c + d*x)*sqrt(b)*sqrt(a)*sqrt( a - b)*sqrt(2*sqrt(a)*sqrt(a - b) + 2*a - b)*atan((e**(c + d*x)*b)/(sqrt(b )*sqrt(2*sqrt(a)*sqrt(a - b) + 2*a - b)))*a*b**2 - 2*e**(c + d*x)*sqrt(b)* sqrt(a)*sqrt(a - b)*sqrt(2*sqrt(a)*sqrt(a - b) + 2*a - b)*atan((e**(c + d* x)*b)/(sqrt(b)*sqrt(2*sqrt(a)*sqrt(a - b) + 2*a - b)))*b**3 - 6*e**(5*c...